1. Technical Field
This invention generally relates to the field of sampling, and more specifically, to methods and apparatus for reconstructing non-uniformly sampled signals and to a computer program product for performing reconstruction.
2. Background Information
Monitoring systems may monitor the current status of signal amplitudes in periodic and equidistant intervals and may be used in numerous technical applications. Analog-digital converters (ADC), weather satellites, temperature sensors or even ECG monitoring are typical examples for such monitoring systems. Furthermore, Fourier techniques such as FFT (Fast Fourier Transform) or the DFT (Discrete Fourier Transform) are widely employed in signal processing and related fields, such as image analysis, speech analysis or spectroscopy to name a few. Fourier techniques may be used analyze the frequencies contained in a sampled signal, and are generally based on equidistant sampled signals. Inadequately reconstructed signals may introduce an error in any following signal processing or further application.
A periodic monitoring of the signals current state, also referred to as “sampling”, that complies with the Nyquist theorem, allows a complete reconstruction of the monitored signal from the discrete sampled data. Though an ideal equidistant sampling period is a basic requirement for a full reconstruction of the sampled signal, it is very difficult to realize in technical applications. Thus, the quality of the reconstructed signal is reduced significantly if no other measures are taken into account.
FIG. 1 illustrates a uniform sampler and quantizer (Q) of an ideal sampling. Here, the value- and time-continuous analog input signal x(t) is sampled with a sample period nT and quantized, resulting in a digital output signal x[n] allowing a full reconstruction of the input signal x(t) from the digital output signal x[n].
FIG. 2 illustrates non-uniformly sampling. The input signal x(t) is sampled and quantized in non-equidistant intervals (nT+Δt[n]), resulting in an output signal x[n]+e[n] consisting of an ideal equidistant sampled signal x][n] and an amplitude error e[n], where e[n] represents the difference in amplitude between the uniformly sampled signal and the realistic non-uniformly sampled signal. Here, Δt represents the time offset from the ideal equidistant sample period nT.
FIG. 3 shows an illustrated example of an ideal equidistant sampling period (e.g. 0, 1T, 2T, 3T . . . ), the realistic non-equidistant sampling period (0+Δt[0]T, 1T+Δt[1]T, 2T+Δt[2]T, 3T+Δt[3]T . . . ) and the resulting amplitude error (e[0], e[1], e[2], e[3] . . . ).
FIG. 4 illustrates a reconstruction for time offsets, solved digitally in conventional methods. Here, a reconstructed signal xr[n] is determined from the non-uniformly sampled signal x[n]+e[n] through the known time offset Δt[n]. Ideally, the reconstructed signal xr[n] is equivalent to the signal x[n] sampled with an equidistant sample period.
Conventional methods for the reconstruction of non-uniformly sampled signals so far use multirate filterbanks, time-varying discrete time FIR filters or monitoring systems that are calibrated individually, making a generalization for any kind of monitoring system very difficult. Hence, using discrete data for the reconstruction of the signal may be promising to find a general solution by means of time discrete signal processing.
As a consequence, conventional methods used so far have disadvantages, such as the necessity for filters or filter arrays, to transform the non-uniformly sampled signal into a uniformly sampled signal, and the requirement of new filters, developed for each sample period due to different time offsets. These disadvantages may increase the complexity of the computation and, ultimately, the time needed for the reconstruction of the signal.